Two-Level Approximation Error-Aware Trust-Region Model Management Numerical Experiments Adaptive Stochastic Collocation for PDE-Constrained Optimization under Uncertainty using Sparse Grids and Model Reduction Matthew J. Zahr Advisor: Charbel Farhat Computational and Mathematical Engineering Stanford University Joint work with: Kevin Carlberg (Sandia CA), Drew Kouri (Sandia NM) SIAM Conference on Uncertainty Quantification MS104: Reduced-Order Modeling in Uncertainty Quantification Lausanne, Switzerland April 7, 2016 Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
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Adaptive Stochastic Collocation for PDE-Constrained ... · Numerical Experiments Proposed Approach Introduce two levels of inexactness to obtain an inexpensive, approximate version
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Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Adaptive Stochastic Collocation for PDE-ConstrainedOptimization under Uncertainty using Sparse Grids and
Model Reduction
Matthew J. Zahr
Advisor: Charbel FarhatComputational and Mathematical Engineering
Stanford University
Joint work with: Kevin Carlberg (Sandia CA), Drew Kouri (Sandia NM)
SIAM Conference on Uncertainty QuantificationMS104: Reduced-Order Modeling in Uncertainty Quantification
Lausanne, SwitzerlandApril 7, 2016
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
r : Rnu × Rnµ × Rnξ → Rnu is the discretized stochastic PDE
J : Rnu × Rnµ × Rnξ → R is a quantity of interest
u ∈ Rnu is the PDE state vector
µ ∈ Rnµ is the vector of (deterministic) optimization parameters
ξ ∈ Rnξ is the vector of stochastic parameters
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Literature Review: Stochastic Optimal Control
Stochastic collocation
Dimension-adaptive sparse grids – globalized with trust-region method[Kouri et al., 2013, Kouri et al., 2014]Generalized polynomial chaos – sequential quadratic programming[Tiesler et al., 2012]
+ Orders of magnitude improvement over isotropic sparse grids- Still requires many PDE solves for even moderate dimensional problems
Model order reduction
Goal-oriented, dimension-adaptive, weighted greedy algorithm for trainingstochastic reduced-order model [Chen and Quarteroni, 2015]Extension to optimal control [Chen and Quarteroni, 2014]
+ Reduction in number of PDE solves at cost of large number of ROM solves- Restriction to offline-online framework may lead to unnecessay PDE solves
and large reduced bases
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Proposed Approach
Introduce two levels of inexactness to obtain an inexpensive, approximate versionof the stochastic optimization problem; manage inexactness with trust-region-like
method
Anisotropic sparse grids used for inexact integration of risk measures
Reduced-order models used for inexact evaluations at collocation nodes
Error indicators introduced to account for both sources of inexactness
Refinement of integral approximation and reduced-order model viadimension-adaptive sparse grids and a greedy method over collocation nodes
Embedded in globally convergent trust-region-like algorithm with a strongconnection to error indicators and refinement mechanism
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Sparse GridsModel Reduction
Anisotropic Sparse Grids [Gerstner and Griebel, 2003]
1D Quadrature Rules: Define the difference operator
∆jk ≡ Ejk − Ej−1
k
where E0k ≡ 0 and Ejk as the level-j 1d quadrature rule for dimension k
Anisotropic Sparse Grid:
EI ≡∑i∈I
∆i11 ⊗ · · · ⊗∆
inξnξ
Forward Neighbors:
N (I) = {k + ej | k ∈ I} \ I
Truncation Error: [Gerstner and Griebel, 2003, Kouri et al., 2013]
E− EI =∑i/∈I
∆i11 ⊗ · · · ⊗∆
inξnξ ≈
∑i∈N (I)
∆i11 ⊗ · · · ⊗∆
inξnξ = EN (I)
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Sparse GridsModel Reduction
Stochastic Collocation via Anisotropic Sparse Grids
Stochastic collocation using anisotropic sparse grid nodes used to approximateintegral with summation
minimizeu∈Rnu , µ∈Rnµ
E[J (u, µ, · )]
subject to r(u, µ, ξ) = 0 ∀ξ ∈ Ξ
⇓
minimizeu∈Rnu , µ∈Rnµ
EI [J (u, µ, · )]
subject to r(u, µ, ξ) = 0 ∀ξ ∈ ΞI
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Sparse GridsModel Reduction
Projection-Based Model Reduction to Reduce PDE Size
Model Order Reduction (MOR) assumption: state vector lies inlow-dimensional subspace
u ≈ Φy∂u
∂µ≈ Φ
∂y
∂µ
where
Φ =[φ1
u · · · φkuu
]∈ Rnu×ku is the reduced basis
y ∈ Rku are the reduced coordinates of unu � ku
Substitute assumption into High-Dimensional Model (HDM),r(u, µ, ξ) = 0, and use a Galerkin projection to obtain the square system
ΦTr(Φy, µ, ξ) = 0
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Error-Based Trust-Region Method Recovers Optimal Control
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
0
1
2
3
z
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
1
1.5
x
u
( ) Determinstic (ξ = 0)
( ) HDM, iso-SG (l = 4)
( ) E[u]( ) E[u]± σ[u]( ) E[u]± 2σ[u]
( ) ROM, aniso-SG
( ) E[Φy]( ) E[Φy]± σ[Φy]( ) E[Φy]± 2σ[Φy]
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Error-Based Trust-Region Method Recovers Optimal Control
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
0
1
2
3
z
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
1
1.5
x
u
( ) Determinstic (ξ = 0)
( ) HDM, iso-SG (l = 4)
( ) E[u]
( ) E[u]± σ[u]( ) E[u]± 2σ[u]
( ) ROM, aniso-SG
( ) E[Φy]( ) E[Φy]± σ[Φy]( ) E[Φy]± 2σ[Φy]
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Error-Based Trust-Region Method Recovers Optimal Control
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
0
1
2
3
z
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
1
1.5
x
u
( ) Determinstic (ξ = 0)
( ) HDM, iso-SG (l = 4)
( ) E[u]( ) E[u]± σ[u]
( ) E[u]± 2σ[u]
( ) ROM, aniso-SG
( ) E[Φy]( ) E[Φy]± σ[Φy]( ) E[Φy]± 2σ[Φy]
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Error-Based Trust-Region Method Recovers Optimal Control
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
0
1
2
3
z
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
1
1.5
x
u
( ) Determinstic (ξ = 0)
( ) HDM, iso-SG (l = 4)
( ) E[u]( ) E[u]± σ[u]( ) E[u]± 2σ[u]
( ) ROM, aniso-SG
( ) E[Φy]( ) E[Φy]± σ[Φy]( ) E[Φy]± 2σ[Φy]
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Error-Based Trust-Region Method Recovers Optimal Control
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
0
1
2
3
z
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
1
1.5
x
u
( ) Determinstic (ξ = 0)
( ) HDM, iso-SG (l = 4)
( ) E[u]( ) E[u]± σ[u]( ) E[u]± 2σ[u]
( ) ROM, aniso-SG
( ) E[Φy]
( ) E[Φy]± σ[Φy]( ) E[Φy]± 2σ[Φy]
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Error-Based Trust-Region Method Recovers Optimal Control
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
0
1
2
3
z
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
1
1.5
x
u
( ) Determinstic (ξ = 0)
( ) HDM, iso-SG (l = 4)
( ) E[u]( ) E[u]± σ[u]( ) E[u]± 2σ[u]
( ) ROM, aniso-SG
( ) E[Φy]( ) E[Φy]± σ[Φy]
( ) E[Φy]± 2σ[Φy]
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Error-Based Trust-Region Method Recovers Optimal Control
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
0
1
2
3
z
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
1
1.5
x
u
( ) Determinstic (ξ = 0)
( ) HDM, iso-SG (l = 4)
( ) E[u]( ) E[u]± σ[u]( ) E[u]± 2σ[u]
( ) ROM, aniso-SG
( ) E[Φy]( ) E[Φy]± σ[Φy]( ) E[Φy]± 2σ[Φy]
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Error-Based Trust-Region Method Requires Very Few HDMQueries to Converges to Optimal Control
Prior to each trust-region subproblem, the model (sparse grid, Ik, and basis, Φk)must be constructed such that error indicators are below a tolerance
Iteration 1
1 2 3 4 5
1
2
3
4
5
i1
i 2
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
ξ1
ξ2
I – N (I) – HDM Sample –
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Error-Based Trust-Region Method Requires Very Few HDMQueries to Converges to Optimal Control
Prior to each trust-region subproblem, the model (sparse grid, Ik, and basis, Φk)must be constructed such that error indicators are below a tolerance
Iteration 2
1 2 3 4 5
1
2
3
4
5
i1
i 2
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
ξ1
ξ2
I – N (I) – HDM Sample –
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Error-Based Trust-Region Method Requires Very Few HDMQueries to Converges to Optimal Control
Prior to each trust-region subproblem, the model (sparse grid, Ik, and basis, Φk)must be constructed such that error indicators are below a tolerance
Iteration 3
1 2 3 4 5
1
2
3
4
5
i1
i 2
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
ξ1
ξ2
I – N (I) – HDM Sample –
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Error-Based Trust-Region Method Requires Very Few HDMQueries to Converges to Optimal Control
Prior to each trust-region subproblem, the model (sparse grid, Ik, and basis, Φk)must be constructed such that error indicators are below a tolerance
Iteration 4
1 2 3 4 5
1
2
3
4
5
i1
i 2
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
ξ1
ξ2
I – N (I) – HDM Sample –
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Error-Based Trust-Region Method Requires Very Few HDMQueries to Converges to Optimal Control
Prior to each trust-region subproblem, the model (sparse grid, Ik, and basis, Φk)must be constructed such that error indicators are below a tolerance
Iteration 5
1 2 3 4 5
1
2
3
4
5
i1
i 2
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
ξ1
ξ2
I – N (I) – HDM Sample –
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Error-Based Trust-Region Method Requires Very Few HDMQueries to Converges to Optimal Control
Prior to each trust-region subproblem, the model (sparse grid, Ik, and basis, Φk)must be constructed such that error indicators are below a tolerance
Iteration 6
1 2 3 4 5
1
2
3
4
5
i1
i 2
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
ξ1
ξ2
I – N (I) – HDM Sample –
Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level ApproximationError-Aware Trust-Region Model Management
Numerical Experiments
Global Convergence of Trust-Region Method
The trust-region method finds a sequence of parameters µk such that the gradientof HDM (||∇J (µ)||) converges to 0 from arbitrary starting point – global